In the country of Genovia, the president wants to ensure that the Monetary Committee can activate a device that opens the country’s safe. The safe system is to be activated by a device that obeys the following rules:
Each member of the Monetary Committee has a button to push.
The vice president or the president has a button to push (at least one of them—or both—have a button to push).
The safe opens only if a combination of the president, the vice president, and a number of the committee members push the button.
Complete the following:
Set the exact constrains of the problem.
Design the safe circuit.
Complete the corresponding truth table.
Explain your rationale on the creation of safe circuit.
Write the corresponding Boolean expression.
Specify the input and output variables and the two states of each.
Input:
p = president’s button (1 = pushed, 0 = not pushed)
vp = vice president’s button ( 1= pushed, 0 not pushed)
x, y, z = Monetary committees’ buttons (1 = pushed, 0 = not pushed)
Output:
f = Safe lock (1 = open, 0 = locked))
Constraints of the problem:
- Each member of the Monetary Committee has a button to push. This means there are at least three committee members, and each member has a corresponding button.
- The vice president or the president has a button to push. This means either the vice president, the president, or both of them have a button.
- The safe opens only if a combination of the president, the vice president, and a number of the committee members push the button. This means that there is a specific combination of buttons being pushed that will open the safe.
Safe circuit design:
To design the safe circuit, we can use logic gates to implement the required combination of inputs. Let's break down the conditions:
1. The safe opens only if a combination of the president, the vice president, and a number of the committee members push the button. This implies the use of AND gates.
2. The vice president or the president has a button to push. This can be achieved using an OR gate.
Based on these conditions, we can design the safe circuit using logic gates as follows:
- Connect the output of the AND gate (representing the combination of committee members' buttons) to one input of another AND gate.
- Connect the output of the OR gate (representing the vice president or the president's button) to the other input of the second AND gate.
- The output of this second AND gate will be connected to the safe lock.
Truth table:
To create the corresponding truth table, we need to list all possible combinations of inputs (p, vp, x, y, z) and their corresponding outputs (f). Since we have 5 inputs, there will be 2^5 = 32 possible combinations.
Here is an example truth table with a few rows:
| p | vp | x | y | z | f |
|---|----|---|---|---|----|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| ... | ... | ... | ... | ... | ... |
Rationale for the creation of the safe circuit:
The safe circuit is designed to meet the given constraints. It ensures that the safe can only be opened if a specific combination of buttons is pressed. By using AND gates to represent the combination of committee members' buttons and an OR gate to represent the vice president or the president's button, we can determine whether the safe should be opened or not.
Boolean expression:
The Boolean expression for the safe circuit can be derived from the safe circuit design we discussed earlier:
f = ( (p AND vp) AND (x OR y OR z) )
Input and output variables:
Input variables:
- p: president's button (0 or 1)
- vp: vice president's button (0 or 1)
- x, y, z: Monetary committee members' buttons (0 or 1)
Output variable:
- f: Safe lock (0 or 1)
States of each variable:
- For each input variable (p, vp, x, y, z) and the output variable (f), the two possible states are 0 (not pushed) and 1 (pushed).